Abatement Distance Source

Overview

Engineering reference data for Abatement Distance Source in acoustics.

Key Formulas

Speed of Sound

$c = \sqrt{\gamma R T}$

Speed of sound in an ideal gas.

Sound Level

$L = 10 \log_{10}(I/I_0)$

Decibel level.

Wavelength

$\lambda = c / f$

Wavelength = speed / frequency.

Variables

Derivation of Sound Pressure

The fundamental relationship for sound pressure (p) from a source is derived from the acoustic power (N) and the geometry of its propagation.

For a spherical wavefront (source in free space), the sound power is distributed over the surface area of a sphere (4πr²). The root-mean-square sound pressure is given by:

$$p = \sqrt{\frac{\rho c N}{4 \pi r^2}}$$

For a half-spherical wavefront (e.g., source on a perfectly reflecting plane like the ground), the power is concentrated over half the sphere's area (2πr²). The pressure increases by a factor of √2:

$$p = \sqrt{\frac{2 \rho c N}{4 \pi r^2}}$$

This leads to the generic formula incorporating the directivity coefficient D:

$$p = \sqrt{\frac{D \rho c N}{4 \pi r^2}}$$

Directivity Coefficient (D)

The directivity coefficient is a critical factor that accounts for how acoustic energy is directed in a real environment, rather than being uniformly distributed.

• D = 1: Corresponds to spherical radiation (source in an open field, away from any reflecting surfaces).
• D = 2: Corresponds to half-spherical radiation (source placed on a large, flat, perfectly reflecting surface).
• D > 2: Can occur in more complex scenarios where sound is focused by multiple reflecting surfaces or due to the inherent directivity of the source itself.

The 6 dB Rule

A key practical outcome of the inverse-square relationship (p² ∝ 1/r² or p ∝ 1/r) is the 6 dB rule:

Doubling the distance (r) from a point source reduces the Sound Pressure Level (L_p) by 6 dB.

This is a direct consequence of the logarithmic nature of the decibel scale and the inverse-square law. A doubling of distance halves the sound pressure (p), and 20 log₁₀(0.5) ≈ -6.02 dB.

References

• Original Source

Standard Atmospheric Reference Values

For practical calculations, standard atmospheric conditions are commonly assumed:

These values correspond to dry air at 20°C and 101.325 kPa. The product ρc is known as the characteristic acoustic impedance of air and appears frequently in acoustic calculations.

Sound Power Level Relationship

The sound pressure level can also be related directly to the sound power level (L_W) by:

$L_p = L_W + 10 \log_{10}\left(\frac{D}{4\pi r^2}\right)$

where $L_W = 10 \log_{10}\left(\frac{N}{N_0}\right)$ and $N_0 = 10^{-12}$ W is the reference sound power.

This form is useful when sound power levels are provided by equipment manufacturers in dB rather than as absolute power values.

Excess Attenuation Term

The quantity $10 \log_{10}\left(\frac{D}{4\pi r^2}\right)$ is sometimes called the excess attenuation or divergence term. It accounts for:

• Geometric spreading of the sound wave (the $4\pi r^2$ denominator)
• Directivity effects of the source (the D coefficient)

Note: Higher directivity coefficients correspond to more concentrated energy, resulting in higher sound pressure levels at a given distance compared to spherical spreading.

Example Calculation

The sound pressure level from a source can be calculated using the derived formula. The following example illustrates the practical application:

Problem: Estimate the sound pressure level 10 meters from a wood planer with a sound power of 0.01 W. Assume direct radiation into a free field (D=2), standard air density (ρ = 1.204 kg/m³), and speed of sound (c = 343 m/s).

Solution: Using the sound pressure level formula: $L_p = 20 \log\left( \frac{1}{r} \sqrt{\frac{D \rho c N}{4 \pi}} \frac{1}{p_{ref}} \right)$

Substituting the given values: $L_p = 20 \log\left( \frac{1}{10\ \text{m}} \sqrt{\frac{2 \cdot 1.204\ \frac{\text{kg}}{\text{m}^3} \cdot 343\ \frac{\text{m}}{\text{s}} \cdot 0.01\ \text{W}}{4 \pi}} \cdot \frac{1}{2 \times 10^{-5}\ \text{Pa}} \right)$

$L_p \approx 71\ \text{dB}$

This result demonstrates how sound power, distance, and directivity combine to determine the resulting sound pressure level at the listener's position.

LaTeX Formula Notation

The key formulas for sound pressure attenuation with distance are presented below in LaTeX notation for precise documentation and computation.

• Spherical distance sound pressure: $p^2 = \frac{\rho c N}{4 \pi r^2}$

• Half-spherical distance sound pressure: $p^2 = \frac{2 \rho c N}{4 \pi r^2}$

• Generic expression with directivity coefficient: $p^2 = \frac{D \rho c N}{4 \pi r^2}$

• Sound Pressure Level (Lp) in decibels: $L_p = 20 \log\left(\frac{p}{p_{\text{ref}}}\right) = 20 \log\left(\frac{1}{r} \sqrt{\frac{D \rho c N}{4 \pi}} / p_{\text{ref}}\right)$

Refer to the Variables section for symbol definitions. Note that doubling the distance reduces $L_p$ by approximately 6 dB, as detailed in the 6 dB Rule section.

Interactive Calculator

Use the calculator below to compute sound pressure level based on the fundamental equations.

Example: Wood Planer Sound Pressure

This example demonstrates applying the formulas to a real-world source.

Problem: A wood planer with an estimated sound power of $0.01\ \text{W}$ is operating in a half-spherical free field (directivity coefficient $D=2$). Calculate the sound pressure level at a distance of $10\ \text{m}$.

Solution: Using the sound pressure level formula (Equation 4) with standard atmospheric density $\rho = 1\ \text{kg/m}^3$ and speed of sound $c = 331.2\ \text{m/s}$:

$$L_p = 20 \log \left( \frac{1}{r} \sqrt{\frac{D \rho c N}{4 \pi}} \cdot \frac{1}{p_{\text{ref}}} \right)$$

Substituting the values:

$$L_p = 20 \log \left( \frac{1}{10} \sqrt{\frac{2 \cdot 1 \cdot 331.2 \cdot 0.01}{4 \pi}} \cdot \frac{1}{2 \times 10^{-5}} \right) \approx 71\ \text{dB}$$

This result shows how the sound pressure level diminishes significantly with distance from the source.